even and odd functions

​Even and Odd functions

​Even Functions

​A function is said to be even if its graph is symmetric with respect to the the y-axis. f(-x)=f(x)

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​Even and odd functions

​Odd Functions

​A function is said to be odd if its graph is symmetric with respect to the origin. f(-x)=-f(x)

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Multiple Choice

An even function is symmetric with respect to the...

• y-axis

• x-axis

• origin

Multiple Choice

An odd function is symmetric with respect to...

• the y-axis

• the x-axis

• the origiin

​Notice that in an even function, the sign of the x coordinate will change, for example the points (2,5) and (-2,5) will appear in the same even function.

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​Notice that in an odd function, the signs of the x-coordinate and y-coordinate will change, for example; the points (2,5) and (-2,-5) are the points of an odd function.

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​Determine whether a function is even or odd graphically

The following graph is even, because is you fold it over the y-axis, the left and right sides will match​

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​Determine whether a function is even or odd graphically

The following graph is odd, because is you fold it over the origin, both parts will match​

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​Determine whether a function is even or odd graphically

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The following graph is neither even nor odd because if you fold it over the origin or the y-axis the points will not match

Multiple Choice

Determine whether the graph is even, odd or neither

• even

• odd

• neither

Multiple Choice

Determine whether the graph is even, odd or neither

• even

• odd

• neither

Multiple Choice

Determine whether the graph is even, odd or neither

• even

• odd

• neither

Multiple Choice

Determine whether the graph is even, odd or neither

• even

• odd

• neither

Multiple Choice

Determine whether the graph is even, odd or neither

• even

• odd

• neither

​How to determine whether a funtion is even or odd analytically

​Ex: Decide whether the function is even, odd or neither

g(x)=x³-x

Solution: Replace x for -x

g(-x)= (-x​)³-(-x)= -x³+x

Now compare the signs of all terms in the new function with the original function, if all signs change, is an ODD function​

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​How to determine whether a funtion is even or odd analytically

​Ex: Decide whether the function is even, odd or neither

g(x)= x⁴+x²+1

Solution: Replace x for -x

g(-x)= (-x​)⁴+ (-x)² + 1= x⁴+x²+1

Now compare the signs of all terms in the new function with the original function, if all signs remain the same, is an EVEN function​

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​How to determine whether a funtion is even or odd analytically

​Ex: Decide whether the function is even, odd or neither

g(x)= x⁴+x³

Solution: Replace x for -x

g(-x)= (-x​)⁴+ (-x)³ = x⁴-x³

Now compare the signs of all terms in the new function with the original function, if only some signs change, NEITHER

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Multiple Choice

Determine whether the function $$f\left(x\right)=x^6-2x^2+3$$  

• even

• odd

• neither

Multiple Choice

Determine whether the function $$f\left(x\right)=x^3-5$$  

• even

• odd

• neither

Multiple Choice

Determine whether the function $$f\left(x\right)=x^3-5x$$  

• even

• odd

• neither

Multiple Choice

Determine whether the function $$f\left(x\right)=x\sqrt[]{1-x^2}$$  

• even

• odd

• neither

Multiple Choice

Determine whether the function $$f\left(t\right)=t^2+2t-3$$  

• even

• odd

• neither

Multiple Choice

Determine whether the function $$f\left(s\right)=4s^{\frac{2}{3}}$$  

• even

• odd

• neither

Open Ended

Write using your own words what you learned today about even and odd functions